variants of the symmetry-based indicator ·...

Variants of the symmetry-based indicatorKen Shiozaki∗

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

August 1, 2019

AbstractThe symmetry-based indicator [H. C. Po, A. Vishwanath, H. Watanabe, Nat. Commun. 8, 50 (2017)]

is a practical tool to diagnose topological materials in the band theory. In this note, we present twodirections to generalize the symmetry-based indicator for other classes of topological materials. The oneis for superconductors. The careful definition of the atomic insulators and the trivial vacuum Hamiltonianyields the symmetry-based indicators specific to superconductors. The other is for ingap boundary states.The quotient of the group of atomic insulators by a subset of atomic insulators such as those localizedat the interior of the unit cell gives us the symmetry-based indicator for detecting ingap corner, hinge,and surface states.

1 Introduction

The concept of the symmetry-based indicator (SI) [1, 2] is summarized as something nontrivial that isdetectable from irreps at high-symmetric points. The SI is a realistic tool to diagnose a nontrivial bandtopology in several points listed below. First, the SI is easy to compute for a given band structure. The SI iscomposed only of the data of the numbers of irreps at high-symmetric points, which is the generalization ofearly works for the Fu-Kane-type formulas. [3, 4, 5, 6] The second issue is that in the presence of magneticspace group symmetry it seems impractical to list all topological invariants in the Brillouin zone (BZ).The computation of the K-group via the Aityah-Hirzebruch spectral sequence showed there exist a lotof topological invariants defined on sub-skeletons in the BZ. [7] The explicit construction of topologicalinvariants is a case-by-case problem for each magnetic space group. See Refs. [8, 9, 10, 7] for some topologicalinvariants beyond the ten-fold classification. [11, 12] The third point is the mismatch in between the topologyin the momentum space and that in the real space. [1, 13] In the real space, the atomic insulators, whichare just occupation states of atomic orbitals, are less nontrivial in the viewpoint of topology, however, in themomentum space, inequivalent atomic insulators are sometimes distinguished by a complicated topologicalinvariant. The SI is designed not to detect atomic insulators.

After the proposal and the enumeration of the SI for 1651 magnetic space groups in spinless and spinfulelectric systems, [1, 2] the explicit expressions of the SIs for for 230 space groups and the implications ofnontrivial values of the SI ware explored, [14, 15, 16, 17], which leads to a catalogue of topological insulators(TIs) and topological semimetals based on a first-principle calculation for the material database. [18, 19, 20]

In this note, we present two directions to extend the SI for other classes of topological materials. Theone is the SIs for superconducting states. We see that the careful definition of topological invariants of the

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Bogoliubov-de Gennes (BdG) Hamiltonian relative to the trivial vacuum BdG Hamiltonian gives us newtypes of SIs. Another direction is the SIs for detecting ingap bound states (,and Andreev bound statesfor superconductors) localized at the boundary of a sample. The emergence of such ingap boundary statesdepends on the choice of the unit cell compatible with the boundary termination. Nevertheless, once we fixa unit cell, there emerges a hierarchy of the set of atomic insulators within the unit cell, giving alternativeSIs associated with a small set of atomic insulators localized at the inertia of the unit cell.

This note is inspired by some prior works. Ono and Watanabe discussed the implication of the SIs ofelectric materials for superconducting BdG Hamiltonians. [21] We show that for BdG Hamiltonians thereemerge SIs to detect TSCs and gapless phases specific to superconductors (SCs). In preparing this work,we became aware of Ref. [22], which also discuss the SI for SCs and gives the Z2d -valued SI for d spacedimensions for odd parity SCs. Benalcazar, Li and Hughes introduced the momentum-space topologicalinvariants for the fractional corner charge in 2d spinless insulators with Cn rotation symmetry. [23] Also,Ref. [24] extended their strategy to spinful electrons.

In this paper, we just argue some routes to generalize the SI. We leave the classification and derivationof the SIs as future works.

The plan of this paper is as follows. In Sec. 2, we give a mathematical formulation of the SI so thatit is easy to see generalizations. The hierarchical structure associated with various definitions of nontrivialtopology of the band structure is presented. In Sec. 3, we give several examples of the SIs beyond electricmaterials. We leave Appendix A for how to compute homomorphisms f : A → B of abelian groups, whichcan be used to make the algorism to compute the classification of SIs for superconductors.

Notations— We use the following notations. µ(p)i : SIs for gapless states in p-cells in the momentum space.

ν(p)i : SIs for pth-order TIs/TSCs. ξ(p)

i : SIs for (p− 1)-dimensional ingap boundary states.

2 Formulation

In this section, we illustrate how the SI is formulated for the band theory. To make our discussion concrete,we consider 2-dimensional systems. The formulation for generic space dimensions is parallel. In Sec. 2.2, weassume for simplicity the abelian group E0,0

1 of topological invariants has no torsion. In cases where E0,01

includes torsion, one can formulate the SI by using the method described in Appendix A.

2.1 Outline

Let E0,01 be the set of abelian groups for the spaces of topological invariants at high-symmetric points in

the Brillouin zone (BZ). 1 Elements of E0,01 are characterized by the set {nj} of topological invariants at

high-symmetric points. nj are typically given by the numbers of irreps at high-symmetric points, and takevalues in Z or Z2. Later, we see the explicit forms of SIs are written as linear combinations of njs.

The concept of the SI would be summarized as “something nontrivial” that is detectable from irreps athigh-symmetric points. We would like to exclude the group X composed of “something trivial” from the

1 In what follows, we borrow the notation from the Aityah-Hirzebruch spectral sequence associated with the filtration of theBZ so that the p-skeletons are given by a cell decomposition of the BZ so that all high-symmetric subspaces are contained in asome cell.

2

group E0,01 . Then, the SI associated with X lives in the quotient group E0,0

1 /X. There are many choices ofthe space X. The best way to consider the structure behind the SI is the sequence of subgroups

0 ⊂ fAId≤0({AId≤0}) ⊂ fAI

d≤1({AId≤1}) ⊂ fAId≤2({AId≤2})

∼= fTId≤0({TId≤0}) ⊂ fTI

d≤1({TId≤1}) ⊂ fTId≤2({TId≤2})

∼= E0,03 ⊂ E0,0

2 ⊂ E0,01 (1)

This is the filtration for 2-spatial dimensions. The ingredients above are introduced in order.

The groups E0,0p (p = 2, 3) are defined as the kernel of the (p− 1)th differential of the Atiyah-Hirzebruch

spectral sequence [7]

E0,0p+1 := Ker [d0,0

p : E0,0p → Ep,−p+1

p ]. (2)

Here, the group Ep,−p+1p is the set of abelian groups generated by gapless Dirac points in the form

∑pj=1 kjγj

within p-cells. The differential d0,0p measures the obstacle to glue the band structrue specified by a data of

E0,0p toghther at the whole p-cells. The first differential d0,0

1 is easily computed by using the irreduciblecharacter, which is called the compatibility relation in the band theory. [25] By design, a band structuren = {nj} ∈ E0,0

1 with n ∈ E0,0p and n /∈ E0,0

p+1 is a (semi)metal whose band gap closes at a p-cell. Anotherviewpoint is the isomorphism E0,0

p /E0,0p+1∼= Im d0,0

p , where d0,0p expresses the creation of gapless points in

p-cells accompanied with the band inversion of the set of irreps in E0,0p .

We define the group {TId≤p}(p = 0, 1, 2) as the abelian group generated by possible TIs/TSCs supportedon d-dimensional subregions in the real space where d is less than or equal to p. In particular, the group{TId≤0} is generated by atomic insulators. Elements of {TId≤p} can be dependent each other, but they shouldbe exhaustive so that it covers all TIs/TSCs with dimension less than or equal to p. The homomorphism

fTId≤p : {TId≤p} → E0,0

1 , x 7→ fTId≤p(x), (3)

is defined as the set of topological invariants at high-symmetric points for the model x ∈ {TId≤p}. Sincex ∈ {TId≤p} represents an insulator/superconductor, there are no gap-closing points in the BZ, implies thatf({TId≤p}) ⊂ E0,0

3 for p ≤ 2. Moreover, f({TId≤2}) ∼= E0,03 holds true, since the set of irreps n ∈ E0,0

3 canbe glued together in the whole BZ, implying that there exists a TI/TSC in the real space.

The definition of the group {AId≤p}(p = 0, 1, 2) is more involved. It relates to the ingap localized stateslocalized at corners, hinges of materials. [26] We define {AId≤2}(∼= {TId≤0}) as the abelian group generatedby all atomic insulators. [1, 13] We may further define subgroups of atomic insulators as follows. We define{AId≤1} as the abelian group generated by atomic insulators whose Wyckoff positions are not located at thecorners of the unit cell. Similarly, the group {AId≤0} is the defined as the abelian group generated by atomicinsulators whose Wyckoff positions are located at the interior of the unit cell. The homomorphism

fAId≤p : {AId≤p} → E0,0

1 , x 7→ fAId≤p(x) (4)

is defined again as the set of topological invariants nj of E0,01 for the model x ∈ {AId≤p}. Then, for example,

a band structure with irreps n so that n ∈ fAId≤2({AId≤2}) and n /∈ fAI

d≤1({AId≤1}) hosts an ingap cornerstate. We should note that the definition of the unit cell is not unique for a given magnetic space group. Itshould be fixed so that it is compatible with the real-space boundary.

2.2 The derivation of the symmetry-based indicators

In this section, we see how the indicator formulas are made provided that the subgroups in (1) of E0,01 are

given. For the purpose to illustrate the formulation, we for simplicity assume E0,01 is a free abelian group so

3

that its subgroups are also free. This assumption simplifies the calculation of the coimage of d0,0p and the

cokernel of fAId≤p, f

TId≤p. The indicator formulas will be obtained recursively.

2.2.1 E0,02 ⊂ E0,0

1

As noted before, we assume E0,01 is free abelian. Let us write

E0,01 =

m⊕j=1

Z[bj ], (5)

E1,01 =

k⊕j=1

Z[ej ]⊕l⊕

j=1Zpj [fj ]. (6)

Here, fjs are generators of torsion. A set of irreps of E0,01 is written as n =

∑nj=1 njbj . The group E1,0

1represents gapless points in 1-cells. The first differential d0,0

1 : E0,01 → E1,0

1 , the compatibility relation, isexpressed by a matrix Md0,0

1as in

d0,01 (b1, . . . , bm) = (e1, . . . , ek; f1, . . . ,fl)Md0,0

1, Md0,0

1=

Ac1...cl

, (7)

A ∈ Matk×m(Z), cj ∈ Mat1×m(Z/pjZ), (j = 1, . . . l). (8)

The group E0,02 = Ker d0,0

1 ⊂ E0,01 and the quotient E0,0

1 /E0,02 in which the SIs live are given as follows (see

Appendix A). First we introduce an integer-valued matrix

Md0,01

=

A Oc1 p1...

. . .cl pl

∈ Mat(k+l)×(m+l)(Z), (9)

with cj 7→ cj ∈ Mat1×m(Z), (j = 1, . . . l), integral lifts of cjs. We compute the Smith normal form (SNF) ofMd0,0

1to get

uMd0,01v =

λ1

. . . Oλq

O O

, (10)

where λi(i = 1, . . . q) are nonnegative integers, and u, v are unimodular matrices. The following linearcombinations of bjs generates Ker d0,0

1 ,

E0,02 = Ker d0,0

1 =⟨{

m∑i=1

bivij

}m+l

j=q+1

⟩. (11)

To find the explicit basis of E0,02 , we introduce the submatrix

vsub :=

v1,q+1 · · · v1,m+l...

...vm,q+1 · · · vm,m+l

∈ Matm×(m+l−q)(Z), (12)

4

and consider the SNF of it,

u(1)vsubv(1) =

[D(1) OO O

], D(1) =

d

(1)1

. . .d

(1)m1

. (13)

With this, define new basis of E0,01 by

(b(1)1 , . . . , b(1)

m ) = (b1, . . . , bm)[u(1)]−1. (14)

We have the desired results

E0,02 =

m1⊕j=1

Z[d(1)j b

(1)i ], (15)

E0,01 /E0,0

2 =m1⊕j=1

Zd

(1)j

[b(1)i ]⊕

m⊕j=m1+1

Z[b(1)i ]. (16)

The explicit formulas of the SIs are read from the set n of irreps of E0,01 in the basis of {b(1)

j }mj=1. The SIsdetecting the quotient E0,0

1 /E0,02 are given as

µ(1)i :=

m∑i=1

[u(1)]ijnj ∈{

Z/d(1)j Z (j = 1, . . . ,m1),

Z (j = m1 + 1, . . . ,m).(17)

Here, d(1)j = 1 just means no indicators exist for such j. Note that µ(1)

i themselves are defined as integervalues, but the indicators take values in Z/d(1)

j Z or Z. A nontrivial value of a SI µ(1)i 6= 0 implies that there

exists a gapless point in a 1-cell.

2.2.2 E0,03 ⊂ E0,0

2

In the same way as before, we have the kernel and the cockernel of the second deferential d0,02 : E0,0

2 → E2,−12 .

We compute the SNF of the submatrix like as in (13) to get unimoduar matrices u(2), v(2) and the diagonalmatrix D(2). Introducing the new basis of E0,0

2 by

(b(2)1 , . . . , b(2)

m1) := (d(1)

1 b(1)1 , . . . , d(1)

m1b(1)m1

)[u(2)]−1 = (b(1)1 , . . . , b(1)

m1)D(1)[u(2)]−1, (18)

we have

E0,03 =

m2⊕j=1

Z[d(2)j b

(2)j ], (19)

E0,02 /E0,0

3 =m2⊕j=1

Zd

(2)j

[b(2)j ]⊕

m1⊕j=m2+1

Z[b(2)j ]. (20)

The second SIs are given as

µ(2)i =

m1∑j=1

[u(2)]ij ×µ

(1)j

d(1)j

, (i = 1, . . . ,m1). (21)

5

Note that the SIs µ(2)i can be fractional numbers if the first SIs µ(1)

i are nonzero. When µ(1)i ≡ 0, namely

µ(1)i /d

(1)i ∈ Z, the second SIs take values in integers as

µ(2)i ∈

{Z/d(2)

j Z (j = 1, . . . ,m2),Z (j = m2 + 1, . . . ,m1).

(22)

A nontrivial value of the second SI µ(2)i 6= 0 implies that there exists a gapless point in the form of 2d Dirac

Hamiltonian k1γ1 + k2γ2 inside a 2-cell.

2.2.3 {TId≤1} → E0,01

The abelian group E0,03 expresses the set of topological numbers at high-symmetric points that can extend

to the entire BZ without a gapless point, i.e., E0,03 represents band insulators/superconductors. The next

step is to derive the SI for 1st-order TIs/TSCs. As noted before, in 2-spatial dimensions, all possibleinsulators/superconductors {TId≤2} cover the group E0,0

3 . Put differently, fTId≤2({TId≤2}) = E0,0

3 .

Therefore, the first SI detecting TIs/TSCs, which we denote by ν(1)i , arises from the subgroup fTI

d≤1({TId≤1}) ⊂E0,0

3 , where {TId≤1} is the abelian group generated by TIs/TSCs supported on 1 dimensional subspaces inthe real-space manifold. 2 Then, by evaluating the topological invariants at high-symmetric points for modelsof {TId≤1}, we have the homomorphism

fTId≤1 : {TId≤1} → E0,0

1 (23)

in the form

f(c1, . . . , cr) = (b1, . . . , bm)MfTId≤1

, (24)

where cj(j = 1, . . . , r) express TIs/TSCs with the dimension less than or equal to 1. Note that the generators{cj}rj=1 include atomic insulators. Since we have assumed E0,0

1 is free, {TId≤1} can be assumed to be a freeabelian group. To derive the SI for the 1st-order TIs/TSCs, we rewrite the homomorphism fTI

d≤1 : {TId≤1} →E0,0

1 in the basis of E0,03 as in

f(c1, . . . , cr) = (d(2)1 b

(2)1 , . . . , d(2)

m2b(2)m2, b

(2)m2+1, . . . , b

(2)m1, b

(1)m1+1, . . . , b

(1)m )MfTI

d≤1, (25)

MfTId≤1

=[

[D(2)]−1

Im−m2

]×[u(2)[D(1)]−1

Im−m1

]× u(1) ×MfTI

d≤1. (26)

Since fTId≤1({TId≤1}) ⊂ E0,0

3 , the representation matrix can be written as

MfTId≤1

=[M subfTId≤1

O

], M sub

fTId≤1∈ Matm2×r(Z). (27)

Introducing the SNF of M subfTId≤1

,

u(3)M subfTId≤1

v(3) =[D(3) OO O

], D(3) =

d

(3)1

. . .d

(3)m3

, (28)

2 There is no efficient algorism to get {TId≤p} yet. Here, we simply assume that {TId≤p} is given.

6

we have

f(c1, . . . , cr)v(3) = (b(3)1 , . . . , b(3)

m2, b

(2)m2+1, . . . , b

(2)m1, b

(1)m1+1, . . . , b

(1)m )

d

(3)1

. . . O

d(3)m3

O O

, (29)

with

(b(3)1 , . . . , b(3)

m2) := (b(2)

1 , . . . , b(2)m2

)[u(3)]−1. (30)

The SIs to detect the 1st-order TIs/TSCs are given as

ν(1)i =

m2∑j=1

[u(3)]ij ×µ

(2)j

d(2)j

∈{

Z/d(3)j Z (j = 1, . . . ,m3),

Z (j = m3 + 1, . . . ,m2).(31)

Again, the SI ν(1)i can be fractional if µ(2)

i s have nontrivial. When, µ(2)i take trivial values, a nontrivial SI

ν(1)i 6= 0 implies a 1st-order TI/TSC.

2.2.4 {AId≤p} → E0,01

Let a1, . . . ,ap0 be atomic insulators located at in the inertia of the unit cell, ap0+1, . . . ,ap1 be those locatedat the edge of the unit cell not including the corner, and ap1+1, . . . ,ap2 be those located at the corner of theunit cell, so that they generate the groups of atomic insulators {AId≤0}, {AId≤1} and {AId≤2}, respectively.In the same way as in (28), we have the SNF of the homomorphisms

fAId≤p : {AId≤p} → E0,0

1 , (32)

where fAId≤p is defined by the set of topological invariants of E0,0

1 for a given model x ∈ {AId≤p}.

Let u(4), v(4), D(4) = diag(d(4)1 , . . . , d

(4)m4) be the data of the SNF of fAI

d≤2 as in (28), we have new SIs

ν(2)i =

m3∑j=1

[u(4)]ij ×µ

(3)j

d(3)j

∈{

Z/d(4)j Z (j = 1, . . . ,m4),

Z (j = m4 + 1, . . . ,m3).(33)

Under the condition that the lower SIs ν(1)i , µ

(2)i , µ

(1)i are trivial, a nontrivial value of ν(2)

i means that theband structure is a 2nd-order TI/TSC.

Similarly, let u(5), v(5), D(5) = diag(d(5)1 , . . . , d


d≤1 as in (28), we have theSIs to detect ingap corner states

ξ(1)i =

m4∑j=1

[u(5)]ij ×µ

(4)j

d(4)j

∈{

Z/d(5)j Z (j = 1, . . . ,m5),

Z (j = m5 + 1, . . . ,m4).(34)

At last, let u(6), v(6), D(6) = diag(d(6)1 , . . . , d


d≤0 as in (28), we have theSIs to detect ingap edge states

ξ(2)i =

m5∑j=1

[u(6)]ij ×µ

(5)j

d(5)j

∈{

Z/d(6)j Z (j = 1, . . . ,m6),

Z (j = m6 + 1, . . . ,m5).(35)

7

Table 1: The list of SIs for two spatial dimensions.

SI Range Targetξ

(2)j (j = 1, . . . ,m6) Z/d(6)

j Z Ingap edge statesξ

(2)j (j = m6 + 1, . . . ,m5) Zξ

(1)j (j = 1, . . . ,m5) Z/d(5)

j Z Ingap corner statesξ

(1)j (j = m5 + 1, . . . ,m4) Zν

(2)j (j = 1, . . . ,m4) Z/d(4)

j Z 2st-order TIs/TSCsν

(2)j (j = m4 + 1, . . . ,m3) Zν

(1)j (j = 1, . . . ,m3) Z/d(3)

j Z 1st-order TIs/TSCsν

(1)j (j = m3 + 1, . . . ,m2) Zµ

(2)j (j = 1, . . . ,m2) Z/d(2)

j Z Gapless states in 2-cellsµ

(2)j (j = m2 + 1, . . . ,m1) Zµ

(1)j (j = 1, . . . ,m1) Z/d(1)

j Z Gapless states in 1-cellsµ

(1)j (j = m1 + 1, . . . ,m) Z

SIs for 2-spatial dimensions based on the subgroups (1) are summarized in Table 1.

2.3 Superconductors

2.3.1 Vacuum and triple

For SCs, we should be careful about what a nontrivial Hamiltonian is. Let E be the one-particle Nambu-Hirbert space compsoed of an atomic insulator and its particle-hole pair on which the BdG Hamilotniandefined. The K-group is represented by a Karoubi’s triple [E,H,H0] of Hamiltonians H,H0 that act on thecommon Nambu-Hilbert space E. [27] For SCs, we have a canonical reference Hamiltonian H0: We can setH0 to be the vacuum Hamiltonian for the atomic insulators

H0(k) = ε

2τz (36)

with the positive chemical potential ε > 0, which can also be written as

H0 = ε∑

R,α,j

f†Rαj fRαj (37)

in the many-body Hilbert space, where R, α, j run over all the degrees of freedom, namely the positions ofthe unit cell, Wyckoff poisitoins, and internal degrees of freedom, respectively, and f†Rαj , fRαj are complexfermion creation and annihilation operators. The positive chemical potential means that the ground state ofH0 is the vacuum state |0〉 of the complex fermions. In other words, for the SI, we should define topologicalinvariants at high-symmetric points for a given BdG Hamiltonian H as a relative index of the pair [H,H0]with H0 = τz the reference BdG Hamiltonian.

8

2.3.2 Atomic insulators

We also define atomic insulators for superconductors. They are defined as the atomic insulators in the usualsense. For a given Nambu-Hilbert space E, the atomic insulator is defined as the fully occupied state thatis represented by the BdG Hamiltonian H(k) = ε

2τz with a negative chemical potential ε < 0. Therefore,as an element of the K-group, the atomic insulator for a Nambu-Hilbert space E is given by the triple[E,H = −τz, H0 = τz].

2.3.3 Weak coupling limit

In usual, the superconducting gap function ∆(k) is much smaller than the energy scale of the normal stateh(k). This means that topological invariants of the BdG Hamiltonian

H =(h(k) ∆(k)

∆(k)† −h(−k)T)

(38)

at high-symmetric points, which compose SIs, can be usually computed solely by the normal part h(k)by adiabatically decreasing the gap function ∆(k) to zero. We call this simplification of the topologicalinvariants the weak coupling limit.

3 Some examples

In this section, we illustrate the framework developed in Sec. 2 with several examples. Some parts of thissection are intentionally written long. It is for the purpose to be easily extensible to general magnetic spacegroups.

3.1 1d SCs

Let us start with the simplest example of the SI for superconductors, that is, 1d class D systems with onlytranslation symmetry. The symmetry constraint in the k-space is just the class D PHS

CH(k)C−1 = −H(−k), C = τxK, (39)

where K is the complex conjugation. At two high-symmetric points k = 0, π the effective AZ class are classD, while that for generic points is class A. Thus, we have E0,0

1 = Z2 +Z2 and E1,01 = Z. Here, Z2 is generated

by the triple

[E = C2, C = τxK,H = −τz, H0 = τz], (40)

and characterized by the Z2-quantized Pfaffian

nk = 1π

Arg Pf [Hτx]Pf [H0τx] ∈ {0, 1} (41)

for k = 0, π. In the weak coupling limit, this is rewritten as

nk = 1π

Argdet[h]det[1] = N [h] mod 2, (42)

9

where N [H] denotes the number of occupied state of the Hamiltonian H, and h is the normal part of theBdG Hamiltonian H. Similarly, Z is generated by the triple

[E = C, H = −1, H0 = 1] (43)

at a generic point k ∈ (0, π), and characterized by the integer-valued invariant

n0→π = N [H1]−N [H0] ∈ Z. (44)

We also have the particle-hole symmetric pair [E = C, H = 1, H0 = −1] at −k ∈ (−π, 0). The first differentiald0,0

1 : E0,01 → E1,0

1 is trivial 3, we have E0,02 = E0,0

1 .

The group {AId≤1} of atomic insulators is generated by the relative difference between the atomic insu-lator and the vacuum in the one-band system

[E = S1 × C2, C = τxK,H(k) = −τz, H0(k) = τz]. (45)

This generates the group {AId≤1} = Z2. Therefore, there is a nontrivial SI associated with the quotientE0,0

2 /{AId≤1} = Z2 that is characterized by the sum of Z2 invariants of E0,01 at k = 0, π,

ν = n0 − nπ = 1π

Arg Pf [H(k = 0)τx]Pf [H(k = π)τx] ∈ {0, 1}. (46)

This is nothing but the Pfaffian formula for the 1d TSC by Kitaev. [28] In the weak coupling limit, the aboveSI is simplified as

ν = N [h(k = 0)]−N [h(k = π)] mod 2. (47)

This means that if the normal state has an odd number of fermi points in between the momentum k = 0 andk = π and the superconducting order induces a mass gap to the fermi points, the system becomes a TSC.

3.2 1d odd-parity SCs

Let us consider 1d TRS-broken odd-parity SCs. The symmetry constraint is summarized as

P (k)H(k)P (k)−1 = H(−k), P (−k)P (k) = 1, (48)CP (k) = −P (−k)C. (49)

in addition to the PHS (39). Here,

P (k) =(p(k)

−p(−k)∗)τ

(50)

is the inversion operator for the BdG Hamiltonian with p(k) one for the normal state.

At the high-symmetric points k = 0, π, the PHS operator C exchanges two irreps P (k) = ±1, implyingthat the effective AZ classes are class A. The group E0,0

1 is given by E0,01 = Z[b0] ⊕ Z[bπ] where each Z is

generated by the following triple[E = C2, C = τxK,P = τz, H = −τz, H0 = τz

](51)

3 Since no nontrivial homomorphisms exist from a torsion Zn to the free abelian group Z. In fact, the Z topological invariant(44) is zero for the pair (H, H0) of the generator (40).

10

characterized by the Z invariant

nk = N+[H(k)]−N+[H0(k)] ∈ Z, (52)

with N±[H] the number of occupied states of the Hamiltonian H with the positive/negative parity P (k) = ±at the high-symmetric point k ∈ {0, π}. In the weak coupling limit, this is simplified as the difference of thenumbers of positive and negative parity eigenstates in the occupied states,

nk ={N+[h(k)] +N−[−h(k)]

}−{N+[1] +N−[−1]

}= N+[h(k)]−N−[h(k)] ∈ Z. (53)

At a generic point k ∈ (0, π) in the BZ, the little group Gk of symmetry group is Z2 that is generatedby CP (k). This symmetry operator satisfies (CP (k))2 = −1, meaning that the effective AZ class is class C.No stable point nodes exist in the 1-cell. Therefore, E1,0

1 = 0, and we have E0,02 = E0,0

1 .

Let {AId≤1} is the group generated by atomic insulators. We have two atomic insulators a0,a12 located

at Wyckoff positions x = 0, 12 in the unit cell. These are classified by Z as well as that for BdG Hamiltonians

in the k-space at a high-symmetric point, thus, {AId≤1} = Z⊕2. In the k-space, the BdG Hamiltonians andsymmetry operators for these atomic insulators and the corresponding vacuum are written as

a0 = [E = S1 × C2, C = τxK,P (k) = τz, H(k) = −τz, H0(k) = τz], (54)

a12 = [E = S1 × C2, C = τxK,P (k) = τze

−ik, H(k) = −τz, H0(k) = τz]. (55)

For these triples, the Z invariants of E0,01 are computed as follows.

n0 nπ

a0 1 1a

12 1 −1

=: (MfAId≤1

)T . (56)

This gives the homomorphism fAId≤1 : {AId≤1} → E0,0

1 ,

fAId≤1(a0,a

12 ) = (b0, bπ)MfAI

d≤1. (57)

We have the nontrivial quotient E0,01 /Im fAI

d≤1 = Z/2Z characterized by the SI

ν = n0 − nπ ={N+[H(0)]−N+[H(π)]

}−{N+[H0(0)]−N+[H0(π)]

}mod 2. (58)

The SI ν detects the same Kitaev chain phase as in Sec. 3.1. In the weak coupling limit, the SI is simplifiedas the SI (47) without inversion symmetry

ν ={N+[h(0)]−N−[h(0)]

}−{N+[h(π)]−N−[h(π)]

}= N [h(0)]−N [h(π)] mod 2. (59)

A simple example is the spinless p-wave SC (the Kitaev chain)

H(k) = (−t cos k − µ)τz + ∆ sin kτy, C = τxK, P (k) = τz. (60)

In the parameter region |µ| < |t| so that there is a fermi point in k ∈ (0, π), a Majorana zero mode γ appearsat the edge.

In the presence of inversion symmetry, one can further construct the SI for the Andreev bound states.Let {AId≤0} = Z[a0] be the group generated by the atomic insulator at the center of the unit cell. Thehomomorphism fAI

d≤0 : {AId≤0} → E0,01 is given by

fAId≤0(a0) = (b0, bπ)MfAI

d≤0, (MfAI

d≤0)T = n0 nπ

a0 1 1 . (61)

11

We have the quotient Im fAId≤1/Im fd≤0 = Z characterized by the SI

ξ := n0 − nπ

2 ∈ 12Z. (62)

This takes a value in Z provided that the SI ν for the TSC is trivial.

The SI ξ for the Andreev bound state is demonstrated for the Hamiltonian H(k)⊕2m with an even numberof copies of the Kitaev chains (60). The system is trivial as a TSC as the SI ν is, however, the SI ξ for theAndreev bound states takes a nontrivial value ξ = m. In fact, at the edge, there exist 2m Majorana fermionsand they form m complex fermions with finite energy.

3.3 1d even-parity SCs

Let us consider 1d TRS-broken even-parity SCs. We have the same symmetry constraints as (39) and (48),but the different algebra for the inversion and PHS operators

CP (k) = P (−k)C (63)

with

P (k) =(p(k)

p(−k)∗)τ

. (64)

The group E0,01 is given by E0,0

1 = Z2[b0+]⊕Z2[b0

−]⊕Z2[bπ+]⊕Z2[bπ−], where each bk± is generated by thetriple

bk± =[{k} × C2, C = τxK,P = ±, H = −τz, H0 = τz

](65)

that is characterized by the Z2-quantized Pfaffian

nk± = 1π

ArgPf[

1±P (k)2 H(k)τx

]Pf[

1±P (k)2 H0(k)τx

] ∈ {0, 1}, (66)

where 1±P (k)2 , (k = 0, π), is the projector onto the positive/negative parity states. In the weak coupling

limit, nk± is reduced as

nk± = N±[h(k)] mod 2, k ∈ {0, π}. (67)

At a generic point k ∈ (0, π), the little group is Z2 generated by CP (k) with (CP (k))2 = 1. The effectiveAZ class is class D, and we have E1,0

1 = Z2[b0→π] with the generator

b0→π =[{k ∈ (0, π)} × C2, CP (k) = τxK,H = −τz, H0 = τz

](68)

characterized by the Pfaffian

n0→π = 1π

Arg Pf [H(k)τxP (k)∗]Pf [H0(k)τxP (k)∗] ∈ {0, 1}, k ∈ (0, π), (69)

12

and

n0→π = N [h(k)] mod 2, k ∈ (0, π), (70)

in the weak coupling limit.

The compatibility relation, which is the first differential d0,01 : E0,0

1 → E1,01 , is given by

d0,01 (b0

+, b0−, b

π+, b

π−) = b0→π [ 1 1 1 1

]. (71)

We have a nontrivial quotient E0,01 /E0,0

2 = Z2 charactrized by the SI

µ = n0+ + n0

− + nπ+ + nπ− mod 2, (72)

which is reduced to

µ = N [h(0)]−N [h(π)] mod 2 (73)

in the weak coupling limit. When µ = 1 mod 2, we have a nodal point (Bogoliubov Fermi surface) in a1-cell, irrespective of the gap function ∆(k).

A simple example is the spinless SC

H(k) = (−t cos k − µ)τz + Re ∆(k)τx − Im ∆(k)τy, |µ| < |t|, C = τxK. (74)

When the gap function ∆(k) obeys the even-parity condition ∆(−k) = ∆(k), one can show ∆(k) vanishes,implies that the fermi point is stable.

The next step is to compute the quotient E0,02 /Im fAI

d≤1. It turns out to be zero, which can proven asfollows. One can show that there is no 1d TCs compatible with the inversion symmetry with even-paritygap function, thus, we have {TId≤1} = 0. Therefore, the subgroup of E0,0

2 starts at the group {AId≤1} of 1datomic insulators, and so the homomorphism fAI

d≤1 : {AId≤1} → E0,02 is surjective.

In a similar way to Sec. 3.2, one can construct the SI for Andreev bound states. Let {AId≤0} = Z2[a0+]⊕

Z2[a0−] be the group generated by the atomic insulators localized at the center of the unit cell. Two generators

are given by the triples

a0± = [S1 × C2, C = τxK,P (k) = ±, H(k) = −τz, H0(k) = τz], (75)

and have topological invariants

n0+ n0

− nπ+ nπ−a0

+ 1 0 1 0a0− 0 1 0 1

=: (MfAId≤0

)T . (76)


1 , fAId=0(a0,a

12 ) = (b0

+, b0−, b

π, bπ−)MfAId=0

. We have thenontrivial quotient E0,0

2 /Im fAId≤0 = Z2 with the SI for the Andreev bound state

ξ = n0− + nπ− mod 2. (77)

This is recast as

ξ = N−[h(0)] +N−[h(π)] mod 2 (78)

13

Γ 𝑋

𝑀

𝛼𝛼

𝛼 𝛼

Figure 1: A C4 symmetric 1-skeleton of the 2d BZ.

in the weak coupling limit. Under the assumption of the triviality of the SI µ and that the dof are not locatedat the boundary of the unit cell, ξ 6= 0 implies the existence of an ingap edge state of the BdG Hamiltonian,i.e., the Andreev bound state.

A prime example is the two-orbital SC with an even-parity inter-orbital gap function (equivalently, thespinful SC with an odd-mirror gap function).

H(k) = (−t cos k − µ)τz + ∆ sin kτyσx, C = τxK, P (k) = σz. (79)

At the edge there are two Majoran zero modes γσx=+, γσx=−, and these Majorana fermions form a complexfermion ψ = (γσx=+ + iγσx=−)/2, and it may have a finite energy εψ†ψ depending on the microscopicstructure of the edge.

3.4 2d TR-symmetric spinless systems with C4 rotation symmetry

Let us consider 2d spinless electrons with time-reversal (TR) and C4-rotation symmetry. The relationshipbetween the corner state and the momentum-space invariants in this system was examined in Ref. [23]. Thesymmetry constraint and the algebra among the symmetry operators in the momentum space is summarizedas

TH(k)T−1 = H(−k), T 2 = 1,C4(k)H(kx, ky)C4(k)−1 = H(c4k), C4(c34k)C4(c24k)C4(c4k)C4(k) = 1,TC4(k) = C4(−k)T, (80)

where c4k = (−ky, kx). At generic points in the BZ, the effective AZ class is AI, where effective TRS isthe composition TC2(k) with (TC2(k))2 = 1. The group E0,0

1 is given by E0,01 = Z3 + Z3 + Z2 generated

respectively by irreps at Γ, M , and X points. We employ the 1-skeleton of the BZ as shown in Fig. 1. Thegroup E1,0

1 = Z + Z is composed of irreps at 1-cells Γ→ X and X →M . Let

(bΓ1 , b

Γ−1, b

Γ(i,−i), b

M1 , bM−1, b

M(i,−i), b

X1 , b

X−1) (81)

14

be the basis of E0,01 , where the superscripts and subscripts represent high-symmetric points and irreps,

respectively. Similally, let (cΓ→X , cX→M ) be the basis of E1,01 .

3.4.1 E0,01 → E1,0

1

The compatibility relation define the first differential

d0,01 : E0,0

1 → E1,01 , d0,0

1 (bΓ1 , . . . , b

X−1) = (cΓ→X , cX→M )Md0,0

1, (82)

Md0,01

=bΓ

1 bΓ−1 bΓ

(i,−i) bM1 bM−1 bM(i,−i) bX1 bX−11 1 2 −1 −1 cΓ→X

1−1 −1 −2 1 1 cX→M−1

. (83)

The SNF of Md0,01

is given as

uMd0,01v =

[I2 O

], (84)

where u, v are unimodular. Introduce the submatrix vsub = {vij}1≤i≤8,3≤j≤8 that spans Ker d0,01 . The SNF

of vsub gives the basis of E0,02 and the SIs. We have

u(1)vsubv(1) =

[I6O

], (85)

and E0,02 =

⊕6j=1 Zb

(1)j with (b(1)

1 , . . . , b(1)8 ) = (bΓ

1 , . . . , bX−1)[u(1)]−1,

[u(1)]−1 =

b(1)1 b

(1)2 b

(1)3 b

(1)4 b

(1)5 b

(1)6 b

(1)7 b

(1)8

−2 −1 0 0 1 1 1 0 bΓ1

0 1 0 0 0 0 0 0 bΓ−1

1 0 0 0 0 0 0 0 bΓ(i,−i)

0 0 −1 −2 1 1 0 0 bM10 0 1 0 0 0 0 0 bM−10 0 0 1 0 0 0 0 bM(i,−i)0 0 0 0 1 0 0 0 bX10 0 0 0 0 1 0 1 bX−1

. (86)

We get two Z-valued SIs µ(1)7 , µ

(1)8 to detect the quotient group E0,0

2 /E0,01∼= Zb

(1)7 ⊕ Zb

(1)8 , which are given

as µ(1)i =

∑8j=1[u(1)]ijnj with

u(1) =

nΓ1 nΓ

−1 nΓ(i,−i) nM1 nM−1 nM(i,−i) nX1 nX−1

µ(1)1 0 0 1 0 0 0 0 0µ

(1)2 0 1 0 0 0 0 0 0µ

(1)3 0 0 0 0 1 0 0 0µ

(1)4 0 0 0 0 0 1 0 0µ

(1)5 0 0 0 0 0 0 1 0µ

(1)6 0 0 0 1 1 2 −1 0µ

(1)7 1 1 2 −1 −1 −2 0 0µ

(1)8 0 0 0 −1 −1 −2 1 1

. (87)

A nontrivial value of the SIs (µ(1)7 , µ

(2)8 ) 6= (0, 0) implies the existence of a gapless point in a 1-cell somewhere.

Actually, µ(1)7 6= 0 (µ(2)

8 6= 0) implies the existence of a fermi line along the 1-cell Γ→M (X →M).

15

3.4.2 E0,02 → E2,−1

2

In the 2-cell α (shown in Fig. 1), the 2-component gapless Dirac point is protected by the TC2(k) symmetrywith the quantized π-Berry phase, which means E2,−1

1 = Z2. Moreover, a single Dirac point can not beabsorbed to 1-cells in a TR and C4 symmetric way, therefore the Z2 classification persists, E2,−1

2 = Z2.Therefore, the second differential d0,0

2 : E0,02 → E2,−1

2 can be nontrivial. In fact, one can show that the basesb

(1)2 , b

(1)3 , b

(1)6 create the Dirac point with π-Berry phase. The second differential is given by

d0,02 : E0,0

2 → E2,−12 , d0,0

2 (b(1)1 , . . . , b

(1)6 ) = cα Md0,0

2, (88)

Md0,02

= b(1)1 b

(1)2 b

(1)3 b

(1)4 b

(1)5 b

(1)6

0 1 1 0 0 1 cα, (89)

where we have written the base of E2,−12 by cα. This can be verified as follows. One can show that in the

presence of the C4 symmetry the Berry phase eiγ∂α around the boundary of the 2-cell α shown in Fig. 1 canbe written as [Chen]

eiγ∂α =wGC4

wMC4

wXC2

, (90)

where wPg is the product of eigenvalues of the symmetry operator g at the high-symmetric point P for theoccupied states. This reduces to

eiγ∂α = (−1)nΓ−1+nM−1+nX−1 (91)

by TRS. From Table (87), we find that if either of the bases b(1)2 , b

(1)2 , b

(1)6 changes the occupation number,

the Berry phase eiγ∂α changes by −1.

An alternative brute-force derivation is to classify possible Hamiltonians of the band inversion followedby the creation of a Dirac point for each high-symmetric point. We will describe the detail elsewhere.

According to the strategy in Sec. 2.2, we introduce an integral lift

Md0,027→ Md0,0

2=[

0 1 1 0 0 1 2]∈ Mat1×7(Z). (92)

The SNF of Md0,02

is given by u′Md0,02v′ =

[I1 O

]. The SNF of the submatrix v′sub = {v′}1≤i≤6,2≤j≤7 of

v′ spanning Ker d0,02 is given by

u(2)v′subv(2) =

[I5

2

], (93)

from which, we have the group E0,03 spanned as

E0,03 =

5⊕j=1

Z[b(2)j ]⊕ Z[2b

(2)6 ] (94)

16

with b(2)j = b

(1)i [[u(2)]−1]ij ,

[u(2)]−1 =

b(2)1 b

(2)2 b

(2)3 b

(2)4 b

(2)5 b

(2)6

1 0 0 0 0 0 b(1)1

0 −1 0 0 −1 −1 b(1)2

0 1 0 0 0 0 b(1)3

0 0 1 0 0 0 b(1)4

0 0 0 1 0 0 b(1)5

0 0 0 0 1 0 b(1)6

. (95)

The SI for detecting E0,02 /E0,0

3 = Z2[b(2)6 ] is given by µ(2)

i =∑6j=1[u(2)]ijµ(1)

j ,

u(2) =

µ(1)1 µ

(1)2 µ

(1)3 µ

(1)4 µ

(1)5 µ

(1)6

µ(2)1 1 0 0 0 0 0µ

(2)2 0 0 1 0 0 0µ

(2)3 0 0 0 1 0 0µ

(2)4 0 0 0 0 1 0µ

(2)5 0 0 0 0 0 1µ

(2)6 0 −1 −1 0 0 −1

. (96)

We have the Z2-valued SI

µ(2)6 = −µ(1)

2 − µ(1)3 − µ

(1)6 = −nG−1 − nM−1 − (nM1 + nM−1 + 2nM(i,−i) − nX1 ) mod 2. (97)

Note that modulo Im d0,01 , namely by using µ(1)

8 = −nM1 − nM−1 − 2nM(i,−i) + nX1 + nX−1 = 0, this is recast as(91). Provided that the first SIs are trivial µ(1)

7 = µ(1)8 = 0, µ(2)

6 6= 0 implies the existence of a Dirac pointwith the quantized π-Berry phase in the 2-cell α.

3.4.3 fAId≤2({AId≤2}) ∼= E3

0,0

Since no building-block TIs in 1- and 2-dimensions exist for class AI systems, any elements of E0,03 should be

represented as an abelian sum of atomic insulators. Let us confirm the equivalence Im fAId≤2∼= E3

0,0 directly.The group {AId≤2} is generated by atomic insulators in the unit cell. Let us write the atomic insulatorinduced by the irrep ρ at the coordinate x in the unit cell by ax

ρ . There are eight generators with the dataof topological invariants in the k-space,

nΓ1 nΓ


a(0,0)1 1 0 0 1 0 0 1 0

a(0,0)−1 0 1 0 0 1 0 1 0

a(0,0)(i,−i) 0 0 1 0 0 1 0 2

a( 1

2 ,12 )

1 1 0 0 0 1 0 0 1a

( 12 ,

12 )

−1 0 1 0 1 0 0 0 1a

( 12 ,

12 )

(i,−i) 0 0 1 0 0 1 2 0a

( 12 ,0)

1 1 1 0 0 0 1 1 1a

( 12 ,0)−1 0 0 1 1 1 0 1 1

=: (Mfd≤2)T . (98)

17


1 and that in the basis of E0,03 as in

fAId≤2(a(0,0)

1 , . . . ,a( 1

2 ,0)−1 ) = (bΓ

1 , . . . , bX−1)MfAI

d≤2

= (b(2)1 , . . . , b

(2)5 , 2b

(2)6 , b

(1)7 , b

(1)8 )

[ I512

]u(2)

I2

u(1)MfAId≤2

, (99)

where

[ I512

]u(2)

I2

u(1)MfAId≤2

=

0 0 1 0 0 1 0 10 1 0 1 0 0 0 10 0 1 0 0 1 1 01 1 0 0 0 2 1 10 0 2 1 1 0 1 10 −1 −1 −1 −1 0 −1 −10 0 0 0 0 0 0 00 0 0 0 0 0 0 0

=:[M subfAId≤2

O

]. (100)

This has the SNF M subfAId≤2∼[I6 O

], which means Im fAI

d≤2∼= E0,0

3 .

3.4.4 {AId≤1} → E0,01

A nontrivial inclusion of insulators starts at the atomic insulators not located at the corner of the unit cell,

fAId≤1 : {AId≤1} → E1

0,0. (101)

The group {AId≤1} is generated by atomic insulators at high-symmetric points (0, 0), ( 12 , 0), (0, 1

2 ) in the unitcell. We have five generators with the data of topological invariants in the k-space,

nΓ1 nΓ


a(0,0)1 1 0 0 1 0 0 1 0

a(0,0)−1 0 1 0 0 1 0 1 0

a(0,0)(i,−i) 0 0 1 0 0 1 0 2

a( 1

2 ,0)1 1 1 0 0 0 1 1 1

a( 1

2 ,0)−1 0 0 1 1 1 0 1 1

=: (Mfd≤1)T . (102)

This gives the homomorphism in the basis of E0,03 .

fAId≤1(a(0,0)

1 , . . . ,a( 1

2 ,0)−1 ) = (bΓ

1 , . . . , bX−1)MfAI

d≤1

= (b(2)1 , . . . , b

(2)5 , 2b

(2)5 , b

(1)7 , b

(1)8 )

[ I512

]u(2)

I2

u(1)MfAId≤1

, (103)

18

where

[ I512

]u(2)

I2

u(1)MfAId≤1

=

0 0 1 0 10 1 0 0 10 0 1 1 01 1 0 1 10 0 2 1 10 −1 −1 −1 −10 0 0 0 00 0 0 0 0

=:[M subfAId≤1

O

]. (104)

The SNF of this is given by

u(3)M subfAId≤1

v(3) =[I4 OO O

], (105)

so we introduce the new basis (b(3)1 , . . . , b

(3)6 ) = (b(2)

1 , . . . , b(2)5 , 2b

(2)6 )[u(3)]−1 with

[u(3)]−1 =

b(3)1 b

(3)2 b

(3)3 b

(3)4 b

(3)5 b

(3)6

0 0 1 0 0 0 b(2)1

0 1 0 0 0 0 b(2)2

0 0 1 1 0 0 b(2)3

1 1 0 1 0 0 b(2)4

0 0 2 1 1 0 b(2)5

0 −1 −1 −1 0 1 2b(2)6

. (106)

We find that the image of fAId≤1 ⊂ E

0,03 is spanned as Im fAI

d≤1 =⊕4

j=1 Z[b(3)j ]. The SIs detecting the quotient

group E0,03 /Im fAI

d≤1 = Z[b(3)5 ]⊕ Z[b(3)

6 ] are given by ξ(1)i =

∑6j=1[u(3)]ijµ(2)

j with

u(3) =

µ(2)1 µ

(2)2 µ

(2)3 µ

(2)4 µ

(2)5 µ

(2)6

ξ(1)1 1 −1 −1 1 0 0ξ

(1)2 0 1 0 0 0 0ξ

(1)3 1 0 0 0 0 0ξ

(1)4 −1 0 1 0 0 0ξ

(1)5 −1 0 −1 0 1 0ξ

(1)6 0 1 1 0 0 1

. (107)

Provided that the lower SIs µ(1)7 , µ

(1)8 , µ

(2)6 are trivial, (ξ(1)

5 , ξ(1)6 ) 6= (0, 0) implies the existence of a corner

state. See Sec. 3.4.6 for demonstration.

3.4.5 {AId≤0} → E10,0

The last step is to evaluate the SI for an ingap edge state. Let {AId≤0} be the abelian group generated byatomic insulators at (0, 0) in the unit cell, says, {a(0,0)

1 ,a(0,0)−1 ,a

(0,0)(i,−i)}. The homomorphism

fAId≤0(a(0,0)

1 ,a(0,0)−1 ,a

(0,0)(i,−i)) = (bΓ

1 , . . . , bX−1)MfAI

d≤0(108)

19

Table 2: SIs for spinless electrons with TR and C4 symmetry in 2d

SI Range nΓ1 nΓ

−1 nΓ(i,−i) nM1 nM−1 nM(i,−i) nX1 nX−1 Target

ξ(2)4 Z 0 0 −1 0 0 1 0 0 Ingap edge statesξ

(1)5 Z 0 0 −1 1 1 1 −1 0 Ingap corner statesξ

(1)6 Z 0 − 1

2 0 − 12 0 0 1

2 0µ

(2)6 Z/2Z 0 −1 0 −1 −2 −2 1 0 Gapless states in 2-cellµ

(1)7 Z 1 1 2 −1 −1 −2 0 0 Gapless states in 1-cellµ

(1)8 Z 0 0 0 −1 −1 −2 1 1

is given by

nΓ1 nΓ


a(0,0)1 1 0 0 1 0 0 1 0

a(0,0)−1 0 1 0 0 1 0 1 0

a(0,0)(i,−i) 0 0 1 0 0 1 0 2

=: (Mfd≤0)T . (109)

To compute the SI detecting edge states, we rewrite the homomorphism fAId≤0 in the basis of Im fAI

d≤1 as in

fAId≤0(a(0,0)

1 ,a(0,0)−1 ,a

(0,0)(i,−i)) = (b(3)

1 , . . . , b(3)6 , b

(1)7 , b

(1)8 )

u(3)[I5

12

]u(2)

I2

u(1)MfAId≤0

, (110)

where u(3)[I5

12

]u(2)

I2

u(1)MfAId≤0

=[I3O

]. (111)

Therefore, the basis {b(3)j }3j=1 is already the basis for Im fAI

d≤0. The SIs detecting the quotient groupIm fAI

d≤1/Im fAId≤0 = Z[b(3)

4 ] is given by ξ(2)4 = ξ

(1)4 . Provided that the SIs µ(1)

7 , µ(1)8 , µ

(2)6 , ξ

(1)6 , ξ

(1)5 are trivial,

the nontrivial SI ξ(1)4 6= 0 implies the existence of an edge state without corner states.

Table 2 summarizes the SIs we got in this section.

3.4.6 Some models

In this section, we demonstrate how the SIs gapless phases and ingap boundary states work.

The first example is the spinless Hamiltonian on the square lattice with a nearest neighbor hoping.

H1(k) = − cos kx − cos ky − µ, C4(k) = 1, T = K. (112)

When the chemical potential is set in −2 < µ < 0, a fermi line exists enclosing the Γ point.

The second example is the following tight-binding model with four dof sitting the center of the unitcell. [29, 23] We consider the π-flux loop hopping around the four corners of the unit cell. The Hamiltonian

20

in the k-space is

H2(k) = −(cos kxσx + sin kxσyτz + cos kyσxτx + sin kyσxτy), C4(k) =

0 0 0 11 0 0 00 1 0 00 0 1 0

, T = K.

(113)

This has four flat bands with Bloch energies Ek = 2, 0, 0,−2, of which the Wanner orbitals are located atthe corner of the unit cell. Here, we focus on the lowest energy band Ek = −2 that has the Bloch state|u(k)〉 = 1

2 (1, e−ikx , e−ikx−iky , e−iky ).

The third example is a semimal phase. Consider the following 2 by 2 Hamiltonian

H3(k) = (cos kx − cos ky)σx + (cos kx + cos ky)σz, C4(k) = σz, T = K. (114)

This model has a Dirac point with the π-Berry phase in the quarter of the BZ.

The fourth example is given by stacking two layers of the semimetal phase H3(k). One can induce afinite mass gap to the Dirac points as

H4(k) = (cos kx − cos ky)σx + (cos kx + cos ky)σz + ε sin kx sin kyσyτy, C4(k) = σz, T = K, (115)

with ε a small constant. The occupied states of H4(k) is the direct sum of Chern insulators with C = 2 andC = −2, and is a fragile topological phase since it cannot be represented as a linear combination of atomicinsulators listed in Table (98). Also, we see the occupied states has a corner state. [23]

The values of the SIs for models H1, . . . ,H4 are listed below.

Model nΓ1 nΓ

−1 nΓ(i,−i) nM1 nM−1 nM(i,−i) nX1 nX−1 ξ

(2)4 ξ

(1)5 ξ

(1)6 µ

(2)6 µ

(1)7 µ

(1)8

H1 1 0 0 0 0 0 0 0 0 0 0 0 1 0H2 1 0 0 0 1 0 0 1 0 1 0 0 0 0H3 0 1 0 1 0 0 1 0 0 0 − 1

2 1 0 0H4 0 2 0 2 0 0 2 0 0 0 −1 0 0 0

(116)

We see that the SIs correctly capture gapless states and ingap boundary states.

3.5 3d TR-symmetric odd-parity SCs

The last example is TR-symmetric odd-parity SCs in the 3-dimensional cubic lattice. The symmetry con-straints are summarized as

TH(k)T−1 = H(−k), T 2 = −1,CH(k)C−1 = −H(−k), C2 = 1,P (k)H(k)P (k)−1 = H(−k), P (−k)P (k) = 1,TP (k) = P (−k)T, CP (k) = −P (−k)C. (117)

At eight high-symmetric points in the BZ, the effective AZ class is class AII, so we have E0,01 = Z8 generated

respectively by

bk =[E = C4, T = isyK,C = τxK,P = τz, H = −τz, H0 = τz

](118)

21

characterized by the Z invariant

nk = 12{N+[H(k)]−N+[H0(k)]

}∈ Z (119)

at eight high-symmetric points. Here, N±[H] denotes the number of occupied states of H with the posi-tive/negative eigenstates of P (k), and the prefactor is due to the Kramaers degeneracy. In the weak couplinglimit, nk is simplified to the difference of the numbers of positive and negative parity occupied states

nk = 12{N+[h(k)]−N−[h(k)]

}∈ Z. (120)

At generic points in the BZ, the EAZ is class CII, meaning no stable gapless points in 1,2, and 3-cells.Thus, we have E0,0

4 = E0,03 = E0,0

2 = E0,01 . Every element in E0,0

1 can be represented by a fully gappedsuperconductor.

3.5.1 Building-block states

To compute the homomorphisms fTId≤p : {TId≤p → E0,0

1 , (p = 0, 1, 2), we should list all possible SCs withspace dimension less than p. For inversion symmetry, the layer construction suffices. 4 Let us introducebuilding-block 0d, 1d, 2d, and 3d states as

H0d = −τz, (121)H1d,i(ki) = (− cos ki − µ)τz + sin kiszτx, (−1 < µ < 1), (122)H2d,ij(ki, kj) = (− cos ki − cos kj − µ)τz + sin kiszτx + sin kjτy, (−2 < µ < 0), (123)H3d(kx, ky, kz) = (− cos kx − cos ky − cos kz − µ)τz + sin kxszτx + sin kyτy + sin kzsxτx, (−3 < µ < −1),

(124)

with TRS and PHS operators T = isyK,C = τxK. The group {TId≤p} is generated by the triples of thesebuilding-block states located at either of eight inversion centers relative to the vacuum Hamiltonian H0 = τz.The 0d atomic insulator H0d can be put on either of eight high-symmetric points in the unit cell. The triples[E,H,H0] have the following topological invariants defined by (119).

[MfTId=0

]T =

P (k) H n000 n100 n010 n001 n110 n011 n101 n111

a(0,0,0)0d τz H0d 1 1 1 1 1 1 1 1

a( 1

2 ,0,0)0d e−ikxτz H0d 1 −1 1 1 −1 1 −1 −1

a(0, 12 ,0)0d e−ikyτz H0d 1 1 −1 1 −1 −1 1 −1

a(0,0, 12 )0d e−ikzτz H0d 1 1 1 −1 1 −1 −1 −1

a( 1

2 ,12 ,0)

0d e−i(kx+ky)τz H0d 1 −1 −1 1 1 −1 −1 1a

(0, 12 ,12 )

0d e−i(ky+kz)τz H0d 1 1 −1 −1 −1 1 −1 1a

( 12 ,0,

12 )

0d e−i(kz+kx)τz H0d 1 −1 1 −1 −1 −1 1 1a

( 12 ,

12 ,

12 )

0d e−i(kx+ky+kz)τz H0d 1 −1 −1 −1 1 1 1 −1

. (125)

4This is not always the case. In general, a generating model forms a pattern of local building-block states in the real spacethat may not be defined on a layer.

22

Here the superscript of ax represents the Wyckoff position on which the Hamiltonian is defined, and that ofnlmn represents the high-symmetric points k = π(l,m, n). Similarly, 1d building-blocks are listed as 5

[MfTId=1

]T =

P (k) H n000 n100 n010 n001 n110 n011 n101 n111

a(0,0,0)1d,x τz H1d,x 1 0 1 1 0 1 0 0

a(0, 12 ,0)1d,x e−ikyτz H1d,x 1 0 −1 1 0 −1 0 0

a(0,0, 12 )1d,x e−ikzτz H1d,x 1 0 1 −1 0 −1 0 0

a(0, 12 ,

12 )

1d,x e−i(ky+kz)τz H1d,x 1 0 −1 −1 0 1 0 0a

(0,0,0)1d,y τz H1d,y 1 1 0 1 0 0 1 0

a( 1

2 ,0,0)1d,y e−ikxτz H1d,y 1 −1 0 1 0 0 −1 0

a(0,0, 12 )1d,y e−ikzτz H1d,y 1 1 0 −1 0 0 −1 0

a( 1

2 ,0,12 )

1d,y e−i(kx+kz)τz H1d,y 1 −1 0 −1 0 0 1 0a

(0,0,0)1d,z τz H1d,z 1 1 1 0 1 0 0 0

a( 1

2 ,0,0)1d,z e−ikxτz H1d,z 1 −1 1 0 −1 0 0 0

a(0, 12 ,0)1d,z e−ikyτz H1d,z 1 1 −1 0 −1 0 0 0

a( 1

2 ,12 ,0)

1d,z e−i(kx+ky)τz H1d,z 1 −1 −1 0 1 0 0 0

. (126)

Similarly, we have 2d building blocks

[MfTId=2

]T =

P (k) H n000 n100 n010 n001 n110 n011 n101 n111

a(0,0,0)2d,xy τz H2d,xy 1 0 0 1 0 0 0 0

a(0,0, 12 )2d,xy e−ikzτz H2d,xy 1 0 0 −1 0 0 0 0

a(0,0,0)2d,yz τz H2d,yz 1 1 0 0 0 0 0 0

a( 1

2 ,0,0)2d,yz e−ikxτz H2d,yz 1 −1 0 0 0 0 0 0

a(0,0,0)2d,xz τz H2d,xz 1 0 1 0 0 0 0 0

a(0, 12 ,0)2d,xz e−ikyτz H2d,xz 1 0 −1 0 0 0 0 0

. (127)

Finally, the 3d building block has the data

[MfTId=3

]T =P (k) H n000 n100 n010 n001 n110 n011 n101 n111

a(0,0,0)3d τz H3d 1 0 0 0 0 0 0 0

. (128)

3.5.2 {TId≤3} → E0,01

Let us write the basis of E0,01 by (b000, . . . , b111). Every element of E0,0

1 (= E0,04 ) should be represented as a

fully gapped SC. To see this, consider the homomorohism fTId≤3 : {TId≤3} → E0,0

1 ,

fTId≤3(a0,0,0

1d , . . . ,a(0,0,0)3d ) = (b000, . . . , b111)MfTI

d≤3, (129)

where

MfTId≤3

=[MfTI

d=0MfTI

d=1MfTI

d=2MfTI

d=3

]. (130)

5 Here, it is not needed to add the model H1d,x on the Wyckoff position ( 12 , 0, 0), since the triple [E, H1d,x, H0] of it has the

same topological invariants as that on the Wyckoff position (0, 0, 0).

23

The SNF of MfTId≤3

is given by

MfTId≤3∼[I8 O

]. (131)

This means Im fTId≤3∼= E0,0

1 .

3.5.3 SIs for TSCs

A nontrivial subgroup starts from the homomorphism fTId≤2 : {TId≤2} → E0,0

1 ,

fTId≤2(a(0,0,0)

1d , . . . ,a(0, 12 ,0)2d,xz ) = (b000, . . . , b111)MfTI

d≤2(132)

with

MfTId≤2

=[MfTI

d=0MfTI

d=1MfTI

d=2

]. (133)

The SNF of MfTId≤2

is given by

u(1)MfTId≤2

v(1) =[I7 O

2 O

]. (134)

Thus, we have a Z2-valued SI ν(1)8 detecting the 1st-order TSC E0,0

1 /Im fTId≤2 = Im fTI

d≤3/Im fTId≤2 = Z2. We

summarize the explicit formulas of SIs later.

Next, we shall compute the homomorphism {TId≤1} → E0,01 ,

fTId≤1(a0,0,0

1d , . . . ,a( 1

2 ,12 ,0)

1d,z ) = (b000, . . . , b111)MfTId≤1

(135)

with

MfTId≤1

=[MfTI

d=0MfTI

d=1

]. (136)

The SNF of the homomorphism fTId≤1 in the basis of Im fTI

d≤2 is given by

u(2)[I7

12

]u(1)MfTI

d≤1v(2) =

[I4

2I4 O

]. (137)

Thus, we have four Z2-valued SIs ν(2)5 , . . . , ν

(2)8 detecting 2nd-order TSCs.

At last, we compute the homomorphism {TId=0} → E0,01 ,

fTId=1(a(0,0,0)

1d , . . . ,a( 1

2 ,12 ,

12 )

0d ) = (b000, . . . , b111)MfTId=0

. (138)

The SNF of the homomorphism fTId=0 in the basis of Im fTI

d≤1 is given by

u(3)[I4

12I4

]u(2)

[I7

12

]u(1)MfTI

d=0v(3) =

[1

2I7

]. (139)

Thus, we have seven Z2-valued SIs ν(3)2 , . . . , ν

(3)8 detecting 3nd-order TSCs.

24

The SIs can be written as 6

SI Range n000 n100 n010 n001 n110 n011 n101 n111 Targetν

(3)2 Z/2Z 1 −1 0 0 0 0 0 0 3rd-order TSCν

(3)3 Z/2Z 1 0 −1 0 0 0 0 0ν

(3)4 Z/2Z 1 0 0 −1 0 0 0 0ν

(3)5 Z/2Z 1

2 − 12 − 1

2 0 12 0 0 0

ν(3)6 Z/2Z 1

2 0 − 12 − 1

2 0 12 0 0

ν(3)7 Z/2Z 1

2 − 12 0 − 1

2 0 0 12 0

ν(3)8 Z/2Z 1

4 − 14 − 1

4 − 14

14

14

14 − 1

4ν

(2)5 Z/2Z 1 −1 −1 0 1 0 0 0 2nd-order TSCν

(2)6 Z/2Z 1 0 −1 −1 0 1 0 0ν

(2)7 Z/2Z 1 −1 0 −1 0 0 1 0ν

(2)8 Z/2Z 1

2 − 12 − 1

2 − 12

12

12

12 − 1

2ν

(1)8 Z/2Z 1 −1 −1 −1 1 1 1 −1 1st-order TSC

(140)

Note that the SI ν(1)8 = 2ν(2)

8 = 4ν(3)8 eventually becomes a Z/8Z-valued SI. Similally, the SIs ν(2)

i = 2ν(3)i (i =

5, 6, 7, 8) become Z/Z4-valued SIs.

It should be noted that the same SIs are obtained by the quotient E0,01 /Im fTI

d=0. The SNF of MfTId=0

isgiven as

uMfTId=0

v =

1

2I34I3

8

. (141)

We have the SIs νi =∑8j=1 uijnj with

u =

SI Range n000 n100 n010 n001 n110 n011 n101 n111

ν1 Z/Z 1 0 0 0 0 0 0 0ν2 Z/2Z 1 −1 0 0 0 0 0 0ν3 Z/2Z 1 0 −1 0 0 0 0 0ν4 Z/2Z 1 0 0 −1 0 0 0 0ν5 Z/4Z 1 −1 −1 0 1 0 0 0ν6 Z/4Z 1 0 −1 −1 0 1 0 0ν7 Z/4Z 1 −1 0 −1 0 0 1 0ν8 Z/8Z 1 −1 −1 −1 1 1 1 −1

. (142)

From the expression (120) in the weak coupling limit, the SIs can be written only with the normal state.For instance,

ν8 = 12

[{N+[h(0, 0, 0)]−N+[h(π, 0, 0)]−N+[h(0, π, 0)]−N+[h(0, 0, π)]

+N+[h(π, π, 0)] +N+[h(π, 0, π)] +N+[h(0, π, π)]−N+[h(π, π, π)]}

−{N−[h(0, 0, 0)]−N−[h(π, 0, 0)]−N−[h(0, π, 0)]−N−[h(0, 0, π)]

+N−[h(π, π, 0)] +N−[h(π, 0, π)] +N−[h(0, π, π)]−N−[h(π, π, π)]}].

(143)

6Here we took unimodular transformations to make the formulas of the SIs simple.

25

The essentially same SIs νis were derived in Ref. [22].

In the same way as in Secs. 3.4.4, 3.4.5, taking the quotient by atomic insulators localized at the interiorof the unit cell, one can define the SIs for Andreev bound states. We do not repeat here.

4 Summary

In this note, we depicted two routes for generalizing the SI for electric material in Refs.[Po,Haruki]. Theone is the SIs for superconductors, another one is the SIs for ingap boundary states. To do so, in Sec. 2,we first empathized that there exists a filtration (1) of the group E0,0

1 of topological invariants at high-symmetric points, which is originated from various definitions of nontrivial topology such as semimetalphases, higher-order TIs/TSCs and ingap boundary states. We illustrated how the explicit formulas of theSIs are constructed in Sec. 2, in the cases where the group E0,0

1 is free abelian. For SCs, there emerges anew family of SIs beyond those for electric materials, from the careful definition of what atomic insulatorsand the trivial vacuum Hamiltonian are. We also showed that taking the quotient of the group E0,0

1 by thesubset of atomic insulators such as ones localized at the interior of the unit cell yields the SIs for detectingingap corner, hinge, and boundary states. We demonstrated our framework does work for a few examples inin Sec. 3. We leave the comprehensive classification of the SIs for SCs and ingap boundary states as a futurework.

Acknowledgement— We thank Akira Furusaki, Motoaki Hirayama, Tomoki Ozawa, Masatoshi Sato, LukaTrifunovic, Youichi Yanase, and Tiantian Zhang for helpful discussions. We especially thank YasuhiroIshitsuka for teaching me how to compute Coker f,Ker f, Im f of a homomorphism f of abelian groups. Thiswork was supported by PRESTO, JST (Grant No. JPMJPR18L4).

A On the computation of a homomorphism f : A→ B

Let A,B finitely generated abelian groups, and f : A → B be a homomorphism between them. We shallcompute Ker f, Im f , and Cokerf . The strategy is as follows. [?, ?] Given an arbitrary abelian group A,there exists a free abelian group F and a surjective group homomorphism πA : F → A. The homomorphismf : A → B can be lifted to a homomorphism f : F → G between free abelian groups. The Smith normalform of the representation matrix of f and the inclusions iA : KerπA → F , iB : KerπB → G give us thedesired groups.

Let us denote the bases of A,B by {aj}nj=1, {bj}Mj=1, respectevely, and F =⊕n

j=1 Z[aj ], G =⊕m

j=1 Z[bj ]

26

for the integral lifts. We have the commutative diagram

0 0y yKer f |kerπA −−−−→ KerπA

f |kerπA−−−−−→ KerπB −−−−→ Cokerf |kerπAy iA

y iB

y yKer f −−−−→

⊕j Z[aj ]

f−−−−→⊕

j Z[bj ] −−−−→ Cokerfy πA

y πB

y yKer f −−−−→ A

f−−−−→ B −−−−→ Cokerfy y0 0

. (144)

The homomorphism f and inclusions iA, iB are given as follows. Let us write

A =k⊕j=1

Z[aj ]⊕n⊕

j=k+1Zpj [aj ], A =

k⊕j=1

Z[aj ]⊕n⊕

j=k+1Z[aj ], (145)

B =l⊕

j=1Z[bj ]⊕

m⊕j=l+1

Zqj [bj ], B =l⊕

j=1Z[bj ]⊕

m⊕j=l+1

Z[bj ]. (146)

The inclusions iA, iB are given as

KerπA =n⊕

j=k+1Z[aj ],

iA(ak+1, . . . , an) = (a1, . . . , ak; ak+1, . . . , an)MiA , MiA =

O

pk+1. . .

pn

, (147)

KerπB =m⊕

j=l+1Z[bj ],

iB(bl+1, . . . , bm) = (b1, . . . , bl; bl+1, . . . , bm)MiB , MiB =

O

ql+1. . .

qm

. (148)

The homomorphism f is represented as

f(a1, . . . , ak; ak+1, . . . , an) = (b1, . . . , bl; bl+1, . . . , bm)

M Oαl+1 βk+1,l+1 · · · βk+1,n...

......

αm βm,l+1 · · · βm,n

, (149)

27

where

M ∈ Matl×k(Z),αj ∈ Mat1×k(Zqj ), (j = l + 1, . . .m),βij ∈ Hom(Zpi ,Zqj ) = (q/gcdpi,qj )Z/qjZ(∼= Zgcdpi,qj

), (i = k + 1, . . . ,m, j = l + 1, . . . , n). (150)

The homomorphism f is given by integral lifts of matrix elements

f(a1, . . . , ak; ak+1, . . . , an) = (b1, . . . , bl; bl+1, . . . , bm)Mf ,

Mf =

M O

αl+1 βk+1,l+1 · · · βk+1,n...

......

αm βm,l+1 · · · βm,n

, (151)

where

αj 7→ αj ∈ Mat1×k(Z), (j = l + 1, . . .m),βij 7→ βij ∈ Z, (i = k + 1, . . . ,m, j = l + 1, . . . , n). (152)

A.1 Cokerf

The quotient group Cokerf = B/Im f is given by the quotient group⊕

j Z[bj ]/Im f modulo iB(KerπB).Therefore, Cokerf is written as [?]

Cokerf =⊕j

Z[bj ]/(

Im f + iB(KerπB)), (153)

where X + Y is X ∪ Y us a set. To compute(Im f + iB(KerπB)

), introduce the homomorphism

f ⊕ iB : A⊕KerB → B,

(f ⊕ iB)(a1, . . . , an; bl+1, . . . , bm) = (b1, . . . , bl; bl+1, . . . , bm)Mf⊕iB ,

Mf⊕iB =

M O O

αl+1 βk+1,l+1 · · · βk+1,n ql+1...

......

. . .αm βm,l+1 · · · βm,n qm

. (154)

Applying the Smith decomposition to the matrix Mf⊕iB , we have

uMf⊕iBv =[Dλ OO O

], Dλ =

λ1. . .

λr

, (155)

(f ⊕ iB)(a1, . . . , an; bl+1, . . . , bm)v = (b1, . . . , bl; bl+1, . . . , bm)u−1

λ1

. . . Oλr

O O

, (156)

28

where λi(i = 1, . . . r) are nonnegative integers, and u, v are unimodular matrices. Introducing the new basisof⊕

j Z[bj ] by

(b′1, . . . , b′m) = (b1, . . . , bm)u−1, (157)

we see that

Im f + iB(KerπB) =r⊕j=1

Z[λj b′j ], (158)

and

Coker f ∼=r⊕j=1

Zλj [b′j ]⊕m⊕

j=r+1Z[b′j ]. (159)

A.2 Ker f

The group Ker f can be computed as [?]

Ker f = f−1(iBKerπB)/iAKerπA. (160)

Here note that, from the commutative diagram (144), for iA(x) ∈ iAKerπA we have f(x) = iB(f |KerπA(x)) ∈iB(KerπB), thus iAKerπA ⊂ f−1(iBKerπB).

Let us first compute f−1(iBKerπB). By using the homomorphism f ⊕ iB introduced before, we have

f−1(iBKerπB) = πA

(Ker (f ⊕ iB)

), (161)

where πA : A⊕KerB → A is the projection. Using the Smith normal form (155), Ker (f ⊕ iB) is spanned by

n∑i=1

aivij +m−l∑i=1

bivn+i,j , j = r + 1, . . . , n+m− l. (162)

Therefore, πA(

Ker (f ⊕ iB))is generated by elements

n∑i=1

aivij , j = r + 1, . . . , n+m− l. (163)

Let us write

vsub =

v1,r+1 · · · v1,n+m−l...

...vn,r+1 · · · vn,n+m−l

. (164)

Applying the Smith decomposition to vsub, we have

u′vsubv′ =

[D′ OO O

], D′ =

d′1. . .

d′s

. (165)

29

In the new basis

(a′1, . . . , a′n) := (a1, . . . , an)u′−1 (166)

of A, we have

πA

(Ker (f ⊕ iB)

)=

s⊕j=1

Z[d′j a′j ]. (167)

The quotient

Ker f =s⊕j=1

Z[d′j a′j ]/iAKerπA (168)

is given as follows. We examine the inclusion iA in the basis of {a′j}nj=1,

iA(ak+1, . . . , an) = (a1, . . . , an)MiA = (a′1, . . . , a′n)u′MiA . (169)

Since iAKerπA ⊂⊕s

j=1 Z[d′j a′j ], the r.h.s. is written as

iA(ak+1, . . . , an) = (a1, . . . , an)MiA = (d′1a′1, . . . , d′sa′s; a′s+1, . . . , a′n)[M ′

O

],

M ′ ∈ Mats×(n−k)(Z),[M ′

O

]=[D′−1 OO In−s

]u′MiA . (170)

The final step is to apply the Smith decomposition to M ′. We have

u(1)M ′v(1) =

d

(1)1

. . . O

d(1)s1

O O

, (171)

iA(ak+1, . . . , an)v(1) = (d′1a′1, . . . , d′sa′s)[u(1)]−1

d

(1)1

. . . O

d(1)s1

O O

. (172)

Let us introduce the basis

(a(1)1 , . . . , a(1)

s ) = (d′1a′1, . . . , d′sa′s)[u(1)]−1. (173)

Then, we arrive at

Ker f =s1⊕j=1

Zd

(1)j

[a(1)j ]⊕

s⊕j=s1+1

Z[a(1)j ]. (174)

30

A.3 Im f

The image of f is written as [?]

Im f = Im f/(Im f ∩ iB(KerπB)

)=(Im f + iB(KerπB)

)/iB(KerπB). (175)

The inclusion iB(KerπB) → Im f + iB(KerπB) is computed as follows. In the basis {b′j}mj=1 introduced in(157), iB is written as

iB(bl+1, . . . , bm) = (b′1, . . . , b′m)uMiB

= (λ1b′1, . . . , λr b

′r; b′r+1, . . . , b

′m)[D−1λ OO Im−r

]uMiB

= (λ1b′1, . . . , λr b

′r; b′r+1, . . . , b

′m)[M ′′

O

], (176)

where M ′′ ∈ Matr×(m−l)(Z). Consider the Smith normal form of M ′′,

u(2)M ′′v(2) =

d

(2)1

. . . O

d(2)r1

O O

. (177)

We have (Im f + iB(KerπB)

)/iBKerπB =

r1⊕j=1

Zd

(2)j

[λj b′j ]⊕r⊕

j=r1+1Z[λj b′j ]. (178)

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